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Choptuik’s Critical Spacetime Exists

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Abstract

About 25 years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, discretely self-similar solution at the threshold of black hole formation. We prove rigorously the existence of a real analytic solution, which we interpret as the solution observed by Choptuik. Using a Fourier-Chebyshev expansion, we find that an important linear differential operator has compact inverse. This is reminiscent of an elliptic operator and explains the real analyticity. Our construction covers an open neighborhood of the past light cone of the singularity. The proof is computer-assisted. Starting from an explicit approximate solution, we show that nearby there is a true solution. We include the complete C source code and a high precision data file with about 80 significant decimal digits (Online Resource), with rigorous error bounds. All computer calculations use integer arithmetic only. We do not study perturbations.

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Acknowledgements

Part of this work was carried out while the authors were visiting NYU Abu Dhabi, in fall 2011. We thank the three anonymous referees for useful suggestions that we have tried to incorporate in the final version of this manuscript.

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  1. Michael Reiterer

    Present address: Hebrew University of Jerusalem, Jerusalem, Israel

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  1. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Michael Reiterer & Eugene Trubowitz

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  1. Michael Reiterer
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  2. Eugene Trubowitz
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Correspondence to Michael Reiterer.

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Communicated by P. Chrusciel

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Reiterer, M., Trubowitz, E. Choptuik’s Critical Spacetime Exists. Commun. Math. Phys. 368, 143–186 (2019). https://doi.org/10.1007/s00220-019-03413-8

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  • DOI: https://doi.org/10.1007/s00220-019-03413-8

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